Single-Qubit Quantum Gates on a Bloch Sphere

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Using the Bloch sphere, a cubit can be represented as a unit vector (shown in red) from the origin to the point on the unit sphere with spherical coordinates
. A single-qubit quantum gate
operating on
produces a rotated qubit
, represented by the green vector. Check the box for "add gate 2?" to perform a second operation using gate
. This produces another qubit
, which is represented by the blue vector. You can choose from the gates H, X, Y, Z, S and T as defined in the Details.
Contributed by: S. M. Blinder (August 2017)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The action of the single-qubit quantum gates can be represented by unitary matrices acting on the qubit
:
Hadamard gate: TextData[""]=(111-1),
Pauli X gate:
InlineMath=(0110),
Pauli Y gate: TextData[""]=(0-ii0),
Pauli Z gate:
InlineMath=(100-1),
phase (or
gate:
InlineMath=(100i),
gate:
InlineMath=(100ei π4).
In general, the operation
InlineMath (αβ) gives a qubit of the form
,
with an overall phase factor of no physical significance. Multiplication by
removes this factor and reduces the qubit to the canonical form
.
References
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge: Cambridge University Press, 2010.
[2] G. Fano and S. M. Blinder, Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers, Berlin: Springer, 2017.
Permanent Citation